Peter v oneil advanced engineering mathematics, 6th edition thomson (2007)

Such a solution is called the general solution of the differential equation.. Each choice of the constant in the general solution yields a particular solution.. This general solution is

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Contents

PART 1 Ordinary Differential Equations 1

Chapter 1 First-Order Differential Equations 3

1.1 Preliminary Concepts 31.1.1 General and Particular Solutions 31.1.2 Implicitly Defined Solutions 41.1.3 Integral Curves 5

1.1.4 The Initial Value Problem 61.1.5 Direction Fields 7

1.2 Separable Equations 111.2.1 Some Applications of Separable Differential Equations 141.3 Linear Differential Equations 22

1.4 Exact Differential Equations 261.5 Integrating Factors 33

1.5.1 Separable Equations and Integrating Factors 371.5.2 Linear Equations and Integrating Factors 371.6 Homogeneous, Bernoulli, and Riccati Equations 381.6.1 Homogeneous Differential Equations 381.6.2 The Bernoulli Equation 42

1.6.3 The Riccati Equation 431.7 Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories 461.7.1 Mechanics 46

1.7.2 Electrical Circuits 511.7.3 Orthogonal Trajectories 531.8 Existence and Uniqueness for Solutions of Initial Value Problems 58Chapter 2 Second-Order Differential Equations 61

2.1 Preliminary Concepts 612.2 Theory of Solutions ofy

+ p
xy
+ q
xy = f
x 622.2.1 The Homogeneous Equationy

+ p
xy
+ q
x = 0 642.2.2 The Nonhomogeneous Equationy

+ p
xy
+ q
xy = f
x 682.3 Reduction of Order 69

2.4 The Constant Coefficient Homogeneous Linear Equation 732.4.1 Case 1: A2 − 4B > 0 73

2.4.2 Case 2: A2 − 4B = 0 74

2.4.3 Case 3: A2− 4B < 0 742.4.4 An Alternative General Solution in the Complex Root Case 752.5 Euler’s Equation 78

2.6 The Nonhomogeneous Equationy

+ p
xy
+ q
xy = f
x 822.6.1 The Method of Variation of Parameters 82

2.6.2 The Method of Undetermined Coefficients 852.6.3 The Principle of Superposition 91

2.6.4 Higher-Order Differential Equations 912.7 Application of Second-Order Differential Equations to a Mechanical System 932.7.1 Unforced Motion 95

2.7.2 Forced Motion 982.7.3 Resonance 1002.7.4 Beats 1022.7.5 Analogy with an Electrical Circuit 103

3.1 Definition and Basic Properties 1073.2 Solution of Initial Value Problems Using the Laplace Transform 1163.3 Shifting Theorems and the Heaviside Function 120

3.3.1 The First Shifting Theorem 1203.3.2 The Heaviside Function and Pulses 1223.3.3 The Second Shifting Theorem 1253.3.4 Analysis of Electrical Circuits 1293.4 Convolution 134

3.5 Unit Impulses and the Dirac Delta Function 1393.6 Laplace Transform Solution of Systems 1443.7 Differential Equations with Polynomial Coefficients 150

Chapter 4 Series Solutions 155

4.1 Power Series Solutions of Initial Value Problems 1564.2 Power Series Solutions Using Recurrence Relations 1614.3 Singular Points and the Method of Frobenius 1664.4 Second Solutions and Logarithm Factors 173

Chapter 5 Numerical Approximation of Solutions 181

5.1 Euler’s Method 1825.1.1 A Problem in Radioactive Waste Disposal 1875.2 One-Step Methods 190

5.2.1 The Second-Order Taylor Method 1905.2.2 The Modified Euler Method 1935.2.3 Runge-Kutta Methods 1955.3 Multistep Methods 197

5.3.1 Case 1 r = 0 1985.3.2 Case 2 r = 1 1985.3.3 Case 3 r = 3 1995.3.4 Case 4 r = 4 199

Contents vii

PART 2 Vectors and Linear Algebra 201

Chapter 6 Vectors and Vector Spaces 203

6.1 The Algebra and Geometry of Vectors 2036.2 The Dot Product 211

6.3 The Cross Product 2176.4 The Vector SpaceRn 2236.5 Linear Independence, Spanning Sets, and Dimension inRn 228

Chapter 7 Matrices and Systems of Linear Equations 237

7.1 Matrices 2387.1.1 Matrix Algebra 2397.1.2 Matrix Notation for Systems of Linear Equations 2427.1.3 Some Special Matrices 243

7.1.4 Another Rationale for the Definition of Matrix Multiplication 2467.1.5 Random Walks in Crystals 247

7.2 Elementary Row Operations and Elementary Matrices 2517.3 The Row Echelon Form of a Matrix 258

7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix 2667.5 Solution of Homogeneous Systems of Linear Equations 272

7.6 The Solution Space of AX = O 280

7.7 Nonhomogeneous Systems of Linear Equations 283

7.7.1 The Structure of Solutions of AX = B 284 7.7.2 Existence and Uniqueness of Solutions of AX = B 285

7.8 Matrix Inverses 293

7.8.1 A Method for Finding A−1 295

8.1 Permutations 2998.2 Definition of the Determinant 3018.3 Properties of Determinants 3038.4 Evaluation of Determinants by Elementary Row and Column Operations 3078.5 Cofactor Expansions 311

8.6 Determinants of Triangular Matrices 3148.7 A Determinant Formula for a Matrix Inverse 3158.8 Cramer’s Rule 318

8.9 The Matrix Tree Theorem 320

Chapter 9 Eigenvalues, Diagonalization, and Special Matrices 323

9.1 Eigenvalues and Eigenvectors 3249.1.1 Gerschgorin’s Theorem 3289.2 Diagonalization of Matrices 3309.3 Orthogonal and Symmetric Matrices 339

9.4 Quadratic Forms 3479.5 Unitary, Hermitian, and Skew Hermitian Matrices 352

PART 3 Systems of Differential Equations and Qualitative Methods 359

Chapter 10 Systems of Linear Differential Equations 361

10.1 Theory of Systems of Linear First-Order Differential Equations 361

10.1.1 Theory of the Homogeneous System X
= AX 365 10.1.2 General Solution of the Nonhomogeneous System X
= AX + G 372 10.2 Solution of X
= AX when A is Constant 374

10.2.1 Solution of X
= AX when A has Complex Eigenvalues 377 10.2.2 Solution of X
= AX when A does not have n Linearly Independent

Eigenvectors 379

10.2.3 Solution of X
= AX by Diagonalizing A 384 10.2.4 Exponential Matrix Solutions of X
= AX 386 10.3 Solution of X
= AX + G 394

10.3.1 Variation of Parameters 394

10.3.2 Solution of X
= AX + G by Diagonalizing A 398

Chapter 11 Qualitative Methods and Systems of Nonlinear Differential Equations 403

11.1 Nonlinear Systems and Existence of Solutions 40311.2 The Phase Plane, Phase Portraits and Direction Fields 40611.3 Phase Portraits of Linear Systems 413

11.4 Critical Points and Stability 42411.5 Almost Linear Systems 43111.6 Lyapunov’s Stability Criteria 45111.7 Limit Cycles and Periodic Solutions 461

PART 4 Vector Analysis 473

Chapter 12 Vector Differential Calculus 475

12.1 Vector Functions of One Variable 47512.2 Velocity, Acceleration, Curvature and Torsion 48112.2.1 Tangential and Normal Components of Acceleration 48812.2.2 Curvature as a Function oft 491

12.2.3 The Frenet Formulas 49212.3 Vector Fields and Streamlines 49312.4 The Gradient Field and Directional Derivatives 49912.4.1 Level Surfaces, Tangent Planes and Normal Lines 50312.5 Divergence and Curl 510

12.5.1 A Physical Interpretation of Divergence 51212.5.2 A Physical Interpretation of Curl 513

Contents ix

Chapter 13 Vector Integral Calculus 517

13.1 Line Integrals 51713.1.1 Line Integral with Respect to Arc Length 52513.2 Green’s Theorem 528

13.2.1 An Extension of Green’s Theorem 53213.3 Independence of Path and Potential Theory in the Plane 53613.3.1 A More Critical Look at Theorem 13.5 539

13.4 Surfaces in 3-Space and Surface Integrals 54513.4.1 Normal Vector to a Surface 54813.4.2 The Tangent Plane to a Surface 55113.4.3 Smooth and Piecewise Smooth Surfaces 55213.4.4 Surface Integrals 553

13.5 Applications of Surface Integrals 55713.5.1 Surface Area 557

13.5.2 Mass and Center of Mass of a Shell 55713.5.3 Flux of a Vector Field Across a Surface 56013.6 Preparation for the Integral Theorems of Gauss and Stokes 56213.7 The Divergence Theorem of Gauss 564

13.7.1 Archimedes’s Principle 56713.7.2 The Heat Equation 56813.7.3 The Divergence Theorem as a Conservation of Mass Principle 57013.8 The Integral Theorem of Stokes 572

13.8.1 An Interpretation of Curl 57613.8.2 Potential Theory in 3-Space 576

PART 5 Fourier Analysis, Orthogonal Expansions, and Wavelets 581

Chapter 14 Fourier Series 583

14.1 Why Fourier Series? 58314.2 The Fourier Series of a Function 58614.2.1 Even and Odd Functions 58914.3 Convergence of Fourier Series 59314.3.1 Convergence at the End Points 59914.3.2 A Second Convergence Theorem 60114.3.3 Partial Sums of Fourier Series 60414.3.4 The Gibbs Phenomenon 60614.4 Fourier Cosine and Sine Series 60914.4.1 The Fourier Cosine Series of a Function 61014.4.2 The Fourier Sine Series of a Function 61214.5 Integration and Differentiation of Fourier Series 61414.6 The Phase Angle Form of a Fourier Series 62314.7 Complex Fourier Series and the Frequency Spectrum 63014.7.1 Review of Complex Numbers 630

14.7.2 Complex Fourier Series 631

Chapter 15 The Fourier Integral and Fourier Transforms 637

15.1 The Fourier Integral 63715.2 Fourier Cosine and Sine Integrals 64015.3 The Complex Fourier Integral and the Fourier Transform 64215.4 Additional Properties and Applications of the Fourier Transform 65215.4.1 The Fourier Transform of a Derivative 652

15.4.2 Frequency Differentiation 65515.4.3 The Fourier Transform of an Integral 65615.4.4 Convolution 657

15.4.5 Filtering and the Dirac Delta Function 66015.4.6 The Windowed Fourier Transform 66115.4.7 The Shannon Sampling Theorem 66515.4.8 Lowpass and Bandpass Filters 66715.5 The Fourier Cosine and Sine Transforms 67015.6 The Finite Fourier Cosine and Sine Transforms 67315.7 The Discrete Fourier Transform 675

15.7.1 Linearity and Periodicity 67815.7.2 The InverseN -Point DFT 67815.7.3 DFT Approximation of Fourier Coefficients 67915.8 Sampled Fourier Series 681

15.8.1 Approximation of a Fourier Transform by anN -Point DFT 68515.8.2 Filtering 689

15.9 The Fast Fourier Transform 69415.9.1 Use of the FFT in Analyzing Power Spectral Densities of Signals 69515.9.2 Filtering Noise From a Signal 696

15.9.3 Analysis of the Tides in Morro Bay 697

Chapter 16 Special Functions, Orthogonal Expansions, and Wavelets 701

16.1 Legendre Polynomials 70116.1.1 A Generating Function for the Legendre Polynomials 70416.1.2 A Recurrence Relation for the Legendre Polynomials 70616.1.3 Orthogonality of the Legendre Polynomials 708

16.1.4 Fourier–Legendre Series 70916.1.5 Computation of Fourier–Legendre Coefficients 71116.1.6 Zeros of the Legendre Polynomials 713

16.1.7 Derivative and Integral Formulas forPn
x 71516.2 Bessel Functions 719

16.2.1 The Gamma Function 71916.2.2 Bessel Functions of the First Kind and Solutions of Bessel’s Equation 72116.2.3 Bessel Functions of the Second Kind 722

16.2.4 Modified Bessel Functions 72516.2.5 Some Applications of Bessel Functions 72716.2.6 A Generating Function forJn
x 73216.2.7 An Integral Formula forJn
x 73316.2.8 A Recurrence Relation forJv
x 73516.2.9 Zeros ofJv
x 737

Contents xi

16.2.10 Fourier–Bessel Expansions 73916.2.11 Fourier–Bessel Coefficients 74116.3 Sturm–Liouville Theory and Eigenfunction Expansions 74516.3.1 The Sturm–Liouville Problem 745

16.3.2 The Sturm–Liouville Theorem 75216.3.3 Eigenfunction Expansions 75516.3.4 Approximation in the Mean and Bessel’s Inequality 75916.3.5 Convergence in the Mean and Parseval’s Theorem 76216.3.6 Completeness of the Eigenfunctions 763

16.4 Wavelets 76516.4.1 The Idea Behind Wavelets 76516.4.2 The Haar Wavelets 76716.4.3 A Wavelet Expansion 77416.4.4 Multiresolution Analysis with Haar Wavelets 77416.4.5 General Construction of Wavelets and Multiresolution Analysis 77516.4.6 Shannon Wavelets 776

PART 6 Partial Differential Equations 779

17.1 The Wave Equation and Initial and Boundary Conditions 78117.2 Fourier Series Solutions of the Wave Equation 786

17.2.1 Vibrating String with Zero Initial Velocity 78617.2.2 Vibrating String with Given Initial Velocity and Zero Initial Displacement 79117.2.3 Vibrating String with Initial Displacement and Velocity 793

17.2.4 Verification of Solutions 79417.2.5 Transformation of Boundary Value Problems Involving the Wave Equation 79617.2.6 Effects of Initial Conditions and Constants on the Motion 798

17.2.7 Numerical Solution of the Wave Equation 80117.3 Wave Motion Along Infinite and Semi-Infinite Strings 80817.3.1 Wave Motion Along an Infinite String 80817.3.2 Wave Motion Along a Semi-Infinite String 81317.3.3 Fourier Transform Solution of Problems on Unbounded Domains 81517.4 Characteristics and d’Alembert’s Solution 822

17.4.1 A Nonhomogeneous Wave Equation 82517.4.2 Forward and Backward Waves 82817.5 Normal Modes of Vibration of a Circular Elastic Membrane 83117.6 Vibrations of a Circular Elastic Membrane, Revisited 83417.7 Vibrations of a Rectangular Membrane 837

18.1 The Heat Equation and Initial and Boundary Conditions 84118.2 Fourier Series Solutions of the Heat Equation 844

18.2.1 Ends of the Bar Kept at Temperature Zero 84418.2.2 Temperature in a Bar with Insulated Ends 84718.2.3 Temperature Distribution in a Bar with Radiating End 84818.2.4 Transformations of Boundary Value Problems Involving the Heat Equation 85118.2.5 A Nonhomogeneous Heat Equation 854

18.2.6 Effects of Boundary Conditions and Constants on Heat Conduction 85718.2.7 Numerical Approximation of Solutions 859

18.3 Heat Conduction in Infinite Media 86518.3.1 Heat Conduction in an Infinite Bar 86518.3.2 Heat Conduction in a Semi-Infinite Bar 86818.3.3 Integral Transform Methods for the Heat Equation in an Infinite Medium 86918.4 Heat Conduction in an Infinite Cylinder 873

18.5 Heat Conduction in a Rectangular Plate 877Chapter 19 The Potential Equation 879

19.1 Harmonic Functions and the Dirichlet Problem 87919.2 Dirichlet Problem for a Rectangle 881

19.3 Dirichlet Problem for a Disk 88319.4 Poisson’s Integral Formula for the Disk 88619.5 Dirichlet Problems in Unbounded Regions 88819.5.1 Dirichlet Problem for the Upper Half Plane 88919.5.2 Dirichlet Problem for the Right Quarter Plane 89119.5.3 An Electrostatic Potential Problem 893

19.6 A Dirichlet Problem for a Cube 89619.7 The Steady-State Heat Equation for a Solid Sphere 89819.8 The Neumann Problem 902

19.8.1 A Neumann Problem for a Rectangle 90419.8.2 A Neumann Problem for a Disk 90619.8.3 A Neumann Problem for the Upper Half Plane 908

PART 7 Complex Analysis 911

Chapter 20 Geometry and Arithmetic of Complex Numbers 913

20.1 Complex Numbers 91320.1.1 The Complex Plane 91420.1.2 Magnitude and Conjugate 91520.1.3 Complex Division 91620.1.4 Inequalities 91720.1.5 Argument and Polar Form of a Complex Number 91820.1.6 Ordering 920

20.2 Loci and Sets of Points in the Complex Plane 92120.2.1 Distance 922

20.2.2 Circles and Disks 92220.2.3 The Equationz − a = z − b 92320.2.4 Other Loci 925

20.2.5 Interior Points, Boundary Points, and Open and Closed Sets 925

Contents xiii

20.2.6 Limit Points 92920.2.7 Complex Sequences 93120.2.8 Subsequences 93420.2.9 Compactness and the Bolzano-Weierstrass Theorem 935

21.1 Limits, Continuity, and Derivatives 93921.1.1 Limits 939

21.1.2 Continuity 94121.1.3 The Derivative of a Complex Function 94321.1.4 The Cauchy–Riemann Equations 94521.2 Power Series 950

21.2.1 Series of Complex Numbers 95121.2.2 Power Series 952

21.3 The Exponential and Trigonometric Functions 95721.4 The Complex Logarithm 966

21.5 Powers 96921.5.1 Integer Powers 96921.5.2 z1 /n for Positive Integern 96921.5.3 Rational Powers 971

21.5.4 Powerszw 972Chapter 22 Complex Integration 975

22.1 Curves in the Plane 97522.2 The Integral of a Complex Function 98022.2.1 The Complex Integral in Terms of Real Integrals 98322.2.2 Properties of Complex Integrals 985

22.2.3 Integrals of Series of Functions 98822.3 Cauchy’s Theorem 990

22.3.1 Proof of Cauchy’s Theorem for a Special Case 99322.4 Consequences of Cauchy’s Theorem 994

22.4.1 Independence of Path 99422.4.2 The Deformation Theorem 99522.4.3 Cauchy’s Integral Formula 99722.4.4 Cauchy’s Integral Formula for Higher Derivatives 100022.4.5 Bounds on Derivatives and Liouville’s Theorem 100122.4.6 An Extended Deformation Theorem 1002

Chapter 23 Series Representations of Functions 1007

23.1 Power Series Representations 100723.1.1 Isolated Zeros and the Identity Theorem 101223.1.2 The Maximum Modulus Theorem 101623.2 The Laurent Expansion 1019

Chapter 24 Singularities and the Residue Theorem 1023

24.1 Singularities 102324.2 The Residue Theorem 103024.3 Some Applications of the Residue Theorem 1037

24.3.1 The Argument Principle 103724.3.2 An Inversion for the Laplace Transform 103924.3.3 Evaluation of Real Integrals 1040

25.1 Functions as Mappings 105525.2 Conformal Mappings 106225.2.1 Linear Fractional Transformations 106425.3 Construction of Conformal Mappings Between Domains 107225.3.1 Schwarz–Christoffel Transformation 1077

25.4 Harmonic Functions and the Dirichlet Problem 108025.4.1 Solution of Dirichlet Problems by Conformal Mapping 108325.5 Complex Function Models of Plane Fluid Flow 1087

PART 8 Probability and Statistics 1097

Chapter 26 Counting and Probability 1099

26.1 The Multiplication Principle 109926.2 Permutations 1102

26.3 Choosing r Objects from n Objects 110426.3.1 r Objects from n Objects, with Order 110426.3.2 r Objects from n Objects, without Order 110626.3.3 Tree Diagrams 1107

26.4 Events and Sample Spaces 111226.5 The Probability of an Event 111626.6 Complementary Events 112126.7 Conditional Probability 112226.8 Independent Events 112626.8.1 The Product Rule 112826.9 Tree Diagrams in Computing Probabilities 113026.10 Bayes’ Theorem 1134

26.11 Expected Value 1139

Chapter 27 Statistics 1143

27.1 Measures of Center and Variation 114327.1.1 Measures of Center 114327.1.2 Measures of Variation 114627.2 Random Variables and Probability Distributions 115027.3 The Binomial and Poisson Distributions 115427.3.1 The Binomial Distribution 115427.3.2 The Poisson Distribution 115727.4 A Coin Tossing Experiment, Normally Distributed Data, and the Bell Curve 115927.4.1 The Standard Bell Curve 1174

27.4.2 The 68, 95, 99.7 Rule 1176

Contents xv

27.5 Sampling Distributions and the Central Limit Theorem 1178

27.6 Confidence Intervals and Estimating Population Proportion 1185

27.7 Estimating Population Mean and the Studentt Distribution 1190

27.8 Correlation and Regression 1194

Answers and Solutions to Selected Problems A1

Index I1

Preface

This Sixth Edition of Advanced Engineering Mathematics maintains the primary goal of

previ-ous editions—to engage much of the post-calculus mathematics needed and used by scientists,engineers, and applied mathematicians, in a setting that is helpful to both students and faculty.The format used throughout begins with the correct developments of concepts such as Fourierseries and integrals, conformal mappings, and special functions These ideas are then brought tobear on applications and models of important phenomena, such as wave and heat propagationand filtering of signals

This edition differs from the previous one primarily in the inclusion of statistics andnumerical methods The statistics part treats random variables, normally distributed data, bellcurves, the binomial, Poisson, and student t-distributions, the central limit theorem, confidenceintervals, correlation, and regression This is preceded by prerequisite topics from probabilityand techniques of enumeration

The numerical methods are applied to initial value problems in ordinary differential tions, including a proposal for radioactive waste disposal, and to boundary value problemsinvolving the heat and wave equations

equa-Finally, in order to include these topics without lengthening the book, some items from thefifth edition have been moved to a website, located at http://engineering.thomsonlearning.com

I hope that this provides convenient accessibility Material selected for this move includessome biographies and historical notes, predator/prey and competing species models, the theoryunderlying the efficiency of the FFT, and some selected examples and problems

The chart on the following page offers a complete organizational overview

Acknowledgments

This book is the result of a team effort involving much more than an author Among those

to whom I owe a debt of appreciation are Chris Carson, Joanne Woods, Hilda Gowans andKamilah Reid-Burrell of Thomson Engineering, and Rose Kernan and the professionals at RPKEditorial Services, Inc I also want to thank Dr Thomas O’Neil of the California PolytechnicState University for material he contributed, and Rich Jones, who had the vision for the firstedition of this book many years ago

Finally, I want to acknowledge the reviewers, whose suggestions for improvements andclarifications are much appreciated:

Preliminary Review

Panagiotis Dimitrakopoulos, University of Maryland

Mohamed M Hafez, University of California, Davis

Jennifer Hopwood, University of Western Australia

Nun Kwan Yip, Purdue University

Eigenfunction Expansions, Completeness

Haar Wavelets

Partial Differential Equations

Qualitative Methods, Stability, Analysis of Critical Points

Vectors, Matrices,

Analysis Systems of

Fourier Series, Integrals TransformsFourier Discrete FourierTransform

Statistics

P r eface xix

Draft Review

Sabri Abou-Ward, University of Toronto

Craig Hildebrand, California State University – Fresno

Seiichi Nomura, University of Texas, Arlington

David L Russell, Virginia Polytechnic Institute and State University

Y.Q Sheng, McMaster University

Peter V O’neil

University of Alabama at Birmingham

P A R T

Ordinary Differential Equations

are differential equations These are ordinary differential equations because they involve only

total derivatives, rather than partial derivatives

Differential equations are interesting and important because they express relationshipsinvolving rates of change Such relationships form the basis for developing ideas and studyingphenomena in the sciences, engineering, economics, and increasingly in other areas, such as thebusiness world and the stock market We will see examples of applications as we learn moreabout differential equations

xy
− y2= ex

is of first order

A solution of a differential equation is any function that satisfies it A solution may be

defined on the entire real line, or on only part of it, often an interval For example,

1.1 Preliminary Concepts

Before developing techniques for solving various kinds of differential equations, we will develop

some terminology and geometric insight

A first-order differential equation is any equation involving a first derivative, but no higher

derivative In its most general form, it has the appearance

Note that y
must be present for an equation to qualify as a first-order differential equation, but

x and/or y need not occur explicitly

A solution of equation (1.1) on an interval I is a function  that satisfies the equation for

all x in I That is,

F
x 
x 

x= 0 for all x in I

For example,


x= 2 + ke−x

is a solution of

y
+ y = 2for all real x, and for any number k Here I can be chosen as the entire real line And


x= x ln
x + cx

is a solution of

y
=y

x+ 1for all x > 0, and for any number c

In both of these examples, the solution contained an arbitrary constant This is a symbolindependent of x and y that can be assigned any numerical value Such a solution is called the

general solution of the differential equation Thus


x= 2 + ke−x

is the general solution of y
+ y = 2

Each choice of the constant in the general solution yields a particular solution For example,

f
x= 2 + e−x g
x= 2 − e−xand

h
x= 2 −√53e−xare all particular solutions of y
+ y = 2, obtained by choosing, respectively, k = 1, −1 and

−√53 in the general solution

Sometimes we can write a solution explicitly giving y as a function of x For example,

y= ke−x

is the general solution of

y
= −y

as can be verified by substitution This general solution is explicit, with y isolated on one side

of an equation, and a function of x on the other

By contrast, consider

y
= − 2xy3+ 23x2y2+ 8e4y

We claim that the general solution is the function y
x implicitly defined by the equation

in which k can be any number To verify this, implicitly differentiate equation (1.2) with respect

to x, remembering that y is a function of x We obtain

2xy3+ 3x2

y2y
+ 2 + 8e4y

y
= 0

and solving for y
yields the differential equation

In this example we are unable to solve equation (1.2) explicitly for y as a function of x,isolating y on one side Equation (1.2), implicitly defining the general solution, was obtained

by a technique we will develop shortly, but this technique cannot guarantee an explicit solution

1.1 Preliminary Concepts 5

A graph of a solution of a first-order differential equation is called an integral curve of the

equation If we know the general solution, we obtain an infinite family of integral curves, onefor each choice of the arbitrary constant

EXAMPLE 1.1

We have seen that the general solution of

y
+ y = 2is

y= 2 + ke−xfor all x The integral curves of y
+ y = 2 are graphs of y = 2 + ke−xfor different choices of

k Some of these are shown in Figure 1.1

x y

20

10

102030

y=1

x
xe

x− ex+ c

for x= 0 Graphs of some of these integral curves, obtained by making choices for c, are shown

EXAMPLE 1.3

The differential equation

y
+ xy = 2has general solution

y
x= e−x 2 /2
x

02e 2 /2d+ ke−x 2 /2

Figure 1.3 shows computer-generated integral curves corresponding to k= 0, 4, 13, −7, −15and−11

The general solution of a first-order differential equation F
x y y
= 0 contains an arbitraryconstant, hence there is an infinite family of integral curves, one for each choice of the constant

If we specify that a solution is to pass through a particular point
x0 y0, then we must find that

particular integral curve (or curves) passing through this point This is called an initial value

problem Thus, a first order initial value problem has the form

F
x y y
= 0 y
x0= y0

in which x and y are given numbers The condition y
x = y is called an initial condition.

FIGURE 1.3 Integral curves of y
+ xy = 2 for k = 0 4 13 −7 −15, and

The effect of the initial condition in this example was to pick out one special integral curve

as the solution sought This suggests that an initial value problem may be expected to have aunique solution We will see later that this is the case, under mild conditions on the coefficients

in the differential equation

Imagine a curve, as in Figure 1.4 If we choose some points on the curve and, at each point,draw a segment of the tangent to the curve there, then these segments give a rough outline ofthe shape of the curve This simple observation is the key to a powerful device for envisioningintegral curves of a differential equation

Here f is a known function Suppose f
x y is defined for all points
x y in some region

R of the plane The slope of the integral curve through a given point
x0 y0 of R is y

x0,which equals f
x0 y0 If we compute f
x y at selected points in R, and draw a small linesegment having slope f
x y at each
x y, we obtain a collection of segments which trace outthe shapes of the integral curves This enables us to obtain important insight into the behavior

of the solutions (such as where solutions are increasing or decreasing, limits they might have

at various points, or behavior as x increases)

A drawing of the plane, with short line segments of slope f
x y drawn at selected points

x y, is called a direction field of the differential equation y
= f
x y The name derives fromthe fact that at each point the line segment gives the direction of the integral curve through that

point The line segments are called lineal elements.

EXAMPLE 1.5

Consider the equation

y
= y2

Here f
x y= y2, so the slope of the integral curve through
x y is y2 Select some points and,through each, draw a short line segment having slope y2 A computer generated direction field

is shown in Figure 1.5(a) The lineal elements form a profile of some integral curves and give

us some insight into the behavior of solutions, at least in this part of the plane Figure 1.5(b)reproduces this direction field, with graphs of the integral curves through
0 1,
0 2,
0 3,

first-S E C T I O N 1 1 PROBLEMS

In each of Problems 1 through 6, determine whether the

given function is a solution of the differential equation

7 y2+ xy − 2x2− 3x − 2y = C

y− 4x − 2 +
x + 2y − 2y
= 0

In each of Problems 12 through 16, solve the initial value

problem and graph the solution Hint: Each of these

dif-ferential equations can be solved by direct integration Use

the initial condition to solve for the constant of integration

In each of Problems 17 through 20 draw some lineal

elements of the differential equation for −4 ≤ x ≤ 4,

−4 ≤ y ≤ 4 Use the resulting direction field to sketch a

graph of the solution of the initial value problem (These

problems can be done by hand.)

DEFINITION 1.1 Separable Differential Equation

A differential equation is called separable if it can be written

y
= A
xB
y

In this event, we can separate the variables and write, in differential form,

1B
y dy= A
x dxwherever B
y= 0 We attempt to integrate this equation, writing

B
y dy=
A
x dx

This yields an equation in x, y, and a constant of integration This equation implicitly definesthe general solution y
x It may or may not be possible to solve explicitly for y
x

y2 dx= e−xdxfor y= 0 Integrate this equation to obtain

y2 In fact, the zero function y
x= 0 is a solution of y
= y2ex, although it cannot be obtainedfrom the general solution by any choice of k For this reason, y
x= 0 is called a singularsolution of this equation

Figure 1.7 shows graphs of particular solutions obtained by choosing k as 0, 3,−3, 6 and

This implicitly defines the general solution In this case, we can solve for y
x explicitly Begin

by taking the exponential of both sides to obtain

in which B is any nonzero number

Now revisit the assumption that x= 0 and y = −1 In the general solution, we actuallyobtain y= −1 if we allow B = 0 Further, the constant function y
x = −1 does satisfy

x2y
= 1 + y Thus, by allowing B to be any number, including 0, the general solution

y
x= −1 + Be−1/xcontains all the solutions we have found In this example, y= −1 is a solution, but not asingular solution, since it occurs as a special case of the general solution

Figure 1.8 shows graphs of solutions corresponding to B= −8 −5 0 4 and 7

y+ 3 lny = 1

3
x− 13−11

3 

Separable equations arise in many contexts, of which we will discuss three

to decrease Assuming (for want of better information) that the victim’s temperature was a

“normal” 986 at the time of death, the lieutenant will try to estimate this time by observing thebody’s current temperature and calculating how long it would have had to lose heat to reachthis point

According to Newton’s law of cooling, the body will radiate heat energy into the room at

a rate proportional to the difference in temperature between the body and the room If T
t isthe body temperature at time t, then for some constant of proportionality k,

T
t= 68 + BektNow the constants k and B must be determined, and this requires information The lieutenantarrived at 9:40 p.m and immediately measured the body temperature, obtaining 944 degrees.Letting 9:40 be time zero for convenience, this means that

e80k=212264so

80k= ln

212264



k= 1

80ln

212264



The lieutenant now has the temperature function:

ln

306264



= t

80ln

212264



Therefore the time of death, according to this mathematical model, was

t=80 ln
306/264

ln
212/264 which is approximately −538 minutes Death occurred approximately 538 minutes before(because of the negative sign) the first measurement at 9:40, which was chosen as time zero.This puts the murder at about 8:46 p.m

1.2 Separable Equations 17

Determination of A and k for a given element requires two measurements Suppose atsome time, designated as time zero, there are M grams present This is called the initial mass.Then

ln



MTM



This gives us k and determines the mass at any time:

m
t= Meln
MT/Mt/T



We obtain a more convenient formula for the mass if we choose the time of the secondmeasurement more carefully Suppose we make the second measurement at that time T= H

at which exactly half of the mass has radiated away At this time, half of the mass remains, so

MT= M/2 and MT/M= 1/2 Now the expression for the mass becomes

m
t= Meln
1/2t/H

or

m
t= Me− ln
2t/H (1.4)This number H is called the half-life of the element Although we took it to be the timeneeded for half of the original amount M to decay, in fact, between any times t1 and t1+ H,exactly half of the mass of the element present at t1will radiate away To see this, write

in an artifact, we can estimate the amount of the decay, and hence the time it took, giving an

estimate of the time the organism was alive This process of estimating the age of an artifact

is called carbon dating Of course, in reality the ratio of14C in the atmosphere has only beenapproximately constant, and in addition a sample may have been contaminated by exposure toother living organisms, or even to the air, so carbon dating is a sensitive process that can lead

to controversial results Nevertheless, when applied rigorously and combined with other testsand information, it has proved a valuable tool in historical and archeological studies

To apply equation (1.4) to carbon dating, use H = 5730 and compute

037= e−0000120968T

We find that

T= − ln
037

0000120968≈ 8,219years This is a little less than one and one-half half-lives, a reasonable estimate if nearly 2

3 ofthe14C has decayed

EXAMPLE 1.13

(Torricelli’s Law) Suppose we want to estimate how long it will take for a container to empty

by discharging fluid through a drain hole This is a simple enough problem for, say, a sodacan, but not quite so easy for a large oil storage tank or chemical facility

We need two principles from physics The first is that the rate of discharge of a fluidflowing through an opening at the bottom of a container is given by

dV

dt = −kAv

in which V
t is the volume of fluid in the container at time t, v
t is the discharge velocity

of fluid through the opening, A is the cross sectional area of the opening (assumed constant),and k is a constant determined by the viscosity of the fluid, the shape of the opening, and thefact that the cross-sectional area of fluid pouring out of the opening is slightly less than that

of the opening itself In practice, k must be determined for the particular fluid, container, andopening, and is a number between 0 and 1

We also need Torricelli’s law, which states that v
t is equal to the velocity of a free-fallingparticle released from a height equal to the depth of the fluid at time t (Free-falling meansthat the particle is influenced by gravity only) Now the work done by gravity in moving theparticle from its initial point by a distance h
t is mgh
t, and this must equal the change inthe kinetic energy,
1

2mv2 Therefore,

v
t=2gh
t

1.2 Separable Equations 19

18

18
h r h

Equation (1.5) contains two unknown functions, V
t and h
t, so one must be eliminated.Let r
t be the radius of the surface of the fluid at time t and consider an interval of timefrom t0to t1= t0+ t The volume V of water draining from the tank in this time equals thevolume of a disk of thickness h (the change in depth) and radius r
t∗, for some t∗ between

t0 and t1 Therefore

so

V t

r2 dh

dt = −kA2gh

Now V has been eliminated, but at the cost of introducing r
t However, from Figure 1.9,

r2= 182−
18 − h2= 36h − h2so

36h− h2 dh

dh= −
08

116

 √

64 dt

so

V t

r2 dh

dt = −kA2gh

Now V has been eliminated, but at the cost of introducing r
t However, from Figure 1.9,

r2=...

36h− h2 dh

dh= −
08

116

 √

64 dt